For two graphs \(G\)  and \(H\) , their lexicographic product \(G \otimes H\)  has the vertex set \(V(G \otimes H)=V(G) \times V(H)\)  in which two vertices \((g_1,h_1)\)  and \((g_2,h_2)\)  are adjacent whenever \(g_1g_2 \in E(G)\)  or \(g_1=g_2\)  and \(h_1h_2 \in E(H)\) . Similarly, the tensor product \(G \times H\)  of two graphs \(G\)  and \(H\)  has the vertex set \(V(G) \times V(H)\)  in which two vertices \((g_1,h_1)\)  and \((g_2,h_2)\)  are adjacent whenever \(g_1g_2 \in E(G)\)  and \(h_1h_2 \in E(H)\) . Clearly, the tensor product is commutative and distributive over edge disjoint union of graphs, i.e, if \(G=G_1 \oplus G_2 \oplus ...\oplus G_{u},\)  then \(G \times H=(G_1 \times H) \oplus ...\oplus (G_u \times H)\) . One can easily observe that \(K_u \otimes \bar{K}_{g} \cong K_{g,g,....g}\) , the complete \(u\) -partite graph in which each partite set has \(g\)  vertices. Hereafter, we denote a complete \(u\) -partite graph with \(g\)  vertices in each partite set as \(K_u \otimes \bar{K}_{g}\) .
It is clear that \((K_u \otimes \bar{K}_{g})-gK_u \cong K_u \times K_g\) , where \(gK_u\)  denotes \(g\)  disjoint copies of \(K_u\) . For more details about product graphs, the readers are referred to [1]}.
